We observe that all the non-zero coefficients of \(|\chi _{1}\rangle \)  lie on
the diagonal of \(\Theta (a,n)\) . This leads to non-vanishing \(\Theta (a,n)\) 
for \(|\chi _{1}\rangle \) . Similary, all the non-zero coefficients of \(|\chi _{2}\rangle \)  lie on the antidiagonal of \(\Theta (a,n)\)  and therefore \(\ \Theta (a,n)\)  does not vanish for \(|\chi _{2}\rangle \) . In light of Eq. (REF ), for \(n>2\) , \(|\chi _{1}\rangle \)  and \(|\chi _{2}\rangle \)  are
both different from the \(|GHZ\rangle \) , \(|W\rangle \)  and Dicke states under
SLOCC. It can be further demonstrated that \(|\chi _{1}\rangle \)  and \(|\chi _{2}\rangle \)  are entangled, and that \(|\chi _{2}\rangle \)  is
equivalent to \(|\chi _{1}\rangle \)  under SLOCC. We exemplify the result for
the case of four qubits. We find that \(|\chi _{1}\rangle =(1/2)\bigl (|0\rangle +|5\rangle +|10\rangle -|15\rangle \bigr )\)  and it was shown in [1]} that \(|\chi _{1}\rangle \)  is different from the \(|GHZ\rangle \) , \(|W\rangle \) , and Dicke states under SLOCC.
